\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  49.0 & -8.00 x_{1} & +  3.00 x_{2} & -9.00 x_{3} & -1.00 x_{4} & -6.00 x_{5} & -2.00 x_{6} & +  6.00 x_{7}\\
 x_{9}   &  0.0 & +  5.00 x_{1} & -1.00 x_{2} &   & +  5.00 x_{4} & -1.00 x_{5} & +  1.00 x_{6} & -1.00 x_{7}\\
 x_{10}   &  -3.0 & -9.00 x_{1} & +  6.00 x_{2} & +  5.00 x_{3} & + 10.00 x_{4} & -3.00 x_{5} & -1.00 x_{6} & +  1.00 x_{7}\\
 x_{11}   &  -1.0 & -1.00 x_{1} & -3.00 x_{2} & +  3.00 x_{3} & -5.00 x_{4} & +  5.00 x_{5} & -6.00 x_{6} & + 10.00 x_{7}\\
 x_{12}   &  0.0 & +  1.00 x_{1} & -4.00 x_{2} & +  8.00 x_{3} &   & +  3.00 x_{5} & -6.00 x_{6} & +  7.00 x_{7}\\
 x_{13}   &  1.0 & -8.00 x_{1} & +  9.00 x_{2} & +  3.00 x_{3} & +  3.00 x_{4} & +  2.00 x_{5} & +  1.00 x_{6} & -7.00 x_{7}\\
 x_{14}   &  -14.0 & -6.00 x_{1} & -5.00 x_{2} & +  7.00 x_{3} &   & +  4.00 x_{5} & +  4.00 x_{6} & +  8.00 x_{7}\\
 x_{15}   &  56.0 & -5.00 x_{1} & -8.00 x_{2} & -10.00 x_{3} & +  7.00 x_{4} & +  3.00 x_{5} & -2.00 x_{6} & -7.00 x_{7}\\
\hline
z    &  0.0 & -5.00 x_{1} & -5.00 x_{2} & -1.00 x_{3} & -3.00 x_{4} & +  1.00 x_{5} & +  5.00 x_{6} & +  4.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  8.00 y_{8} & -5.00 y_{9} & +  9.00 y_{10} & +  1.00 y_{11} & -1.00 y_{12} & +  8.00 y_{13} & +  6.00 y_{14} & +  5.00 y_{15}\\
 y_{2}   &  1.0 & -3.00 y_{8} & +  1.00 y_{9} & -6.00 y_{10} & +  3.00 y_{11} & +  4.00 y_{12} & -9.00 y_{13} & +  5.00 y_{14} & +  8.00 y_{15}\\
 y_{3}   &  1.0 & +  9.00 y_{8} &   & -5.00 y_{10} & -3.00 y_{11} & -8.00 y_{12} & -3.00 y_{13} & -7.00 y_{14} & + 10.00 y_{15}\\
 y_{4}   &  1.0 & +  1.00 y_{8} & -5.00 y_{9} & -10.00 y_{10} & +  5.00 y_{11} &   & -3.00 y_{13} &   & -7.00 y_{15}\\
 y_{5}   &  1.0 & +  6.00 y_{8} & +  1.00 y_{9} & +  3.00 y_{10} & -5.00 y_{11} & -3.00 y_{12} & -2.00 y_{13} & -4.00 y_{14} & -3.00 y_{15}\\
 y_{6}   &  1.0 & +  2.00 y_{8} & -1.00 y_{9} & +  1.00 y_{10} & +  6.00 y_{11} & +  6.00 y_{12} & -1.00 y_{13} & -4.00 y_{14} & +  2.00 y_{15}\\
 y_{7}   &  1.0 & -6.00 y_{8} & +  1.00 y_{9} & -1.00 y_{10} & -10.00 y_{11} & -7.00 y_{12} & +  7.00 y_{13} & -8.00 y_{14} & +  7.00 y_{15}\\
\hline
z    &  -0 & -49.00 y_{8} &   & +  3.00 y_{10} & +  1.00 y_{11} &   & -1.00 y_{13} & + 14.00 y_{14} & -56.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 6 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.14285714286 & -65.71 y_{8} & + 28.29 y_{10} & -34.43 y_{11} & -5.00 y_{7} & +  2.86 y_{12} & + 57.57 y_{13} & +  4.86 y_{3} & -8.57 y_{15}\\
 y_{2}   &  1.85714285714 & + 19.71 y_{8} & -14.29 y_{10} & +  7.43 y_{11} & +  1.00 y_{7} & -3.86 y_{12} & -21.57 y_{13} & -1.86 y_{3} & + 19.57 y_{15}\\
 y_{14}   &  0.142857142857 & +  1.29 y_{8} & -0.71 y_{10} & -0.43 y_{11} & -0.00 y_{7} & -1.14 y_{12} & -0.43 y_{13} & -0.14 y_{3} & +  1.43 y_{15}\\
 y_{9}   &  0.142857142857 & + 16.29 y_{8} & -4.71 y_{10} & +  6.57 y_{11} & +  1.00 y_{7} & -2.14 y_{12} & -10.43 y_{13} & -1.14 y_{3} & +  4.43 y_{15}\\
 y_{5}   &  0.571428571429 & + 17.14 y_{8} & +  1.14 y_{10} & +  3.29 y_{11} & +  1.00 y_{7} & -0.57 y_{12} & -10.71 y_{13} & -0.57 y_{3} & -4.29 y_{15}\\
 y_{6}   &  0.285714285714 & -19.43 y_{8} & +  8.57 y_{10} & +  1.14 y_{11} & -1.00 y_{7} & + 12.71 y_{12} & + 11.14 y_{13} & +  1.71 y_{3} & -8.14 y_{15}\\
 y_{4}   &  0.285714285714 & -80.43 y_{8} & + 13.57 y_{10} & -27.86 y_{11} & -5.00 y_{7} & + 10.71 y_{12} & + 49.14 y_{13} & +  5.71 y_{3} & -29.14 y_{15}\\
\hline
z    &  2.0 & -31.00 y_{8} & -7.00 y_{10} & -5.00 y_{11} & -0.00 y_{7} & -16.00 y_{12} & -7.00 y_{13} & -2.00 y_{3} & -36.00 y_{15}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  31.0 & + 65.71 x_{1} & -19.71 x_{2} & -1.29 x_{14} & -16.29 x_{9} & -17.14 x_{5} & + 19.43 x_{6} & + 80.43 x_{4}\\
 x_{10}   &  7.0 & -28.29 x_{1} & + 14.29 x_{2} & +  0.71 x_{14} & +  4.71 x_{9} & -1.14 x_{5} & -8.57 x_{6} & -13.57 x_{4}\\
 x_{11}   &  5.0 & + 34.43 x_{1} & -7.43 x_{2} & +  0.43 x_{14} & -6.57 x_{9} & -3.29 x_{5} & -1.14 x_{6} & + 27.86 x_{4}\\
 x_{7}   &  6.66133814775e-16 & +  5.00 x_{1} & -1.00 x_{2} & +  0.00 x_{14} & -1.00 x_{9} & -1.00 x_{5} & +  1.00 x_{6} & +  5.00 x_{4}\\
 x_{12}   &  16.0 & -2.86 x_{1} & +  3.86 x_{2} & +  1.14 x_{14} & +  2.14 x_{9} & +  0.57 x_{5} & -12.71 x_{6} & -10.71 x_{4}\\
 x_{13}   &  7.0 & -57.57 x_{1} & + 21.57 x_{2} & +  0.43 x_{14} & + 10.43 x_{9} & + 10.71 x_{5} & -11.14 x_{6} & -49.14 x_{4}\\
 x_{3}   &  2.0 & -4.86 x_{1} & +  1.86 x_{2} & +  0.14 x_{14} & +  1.14 x_{9} & +  0.57 x_{5} & -1.71 x_{6} & -5.71 x_{4}\\
 x_{15}   &  36.0 & +  8.57 x_{1} & -19.57 x_{2} & -1.43 x_{14} & -4.43 x_{9} & +  4.29 x_{5} & +  8.14 x_{6} & + 29.14 x_{4}\\
\hline
z    &  -2.0 & -1.14 x_{1} & -1.86 x_{2} & -0.14 x_{14} & -0.14 x_{9} & -0.57 x_{5} & -0.29 x_{6} & -0.29 x_{4}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  31.0 & + 65.71 x_{1} & -19.71 x_{2} & -1.29 x_{14} & -16.29 x_{9} & -17.14 x_{5} & + 19.43 x_{6} & + 80.43 x_{4}\\
 x_{10}   &  7.0 & -28.29 x_{1} & + 14.29 x_{2} & +  0.71 x_{14} & +  4.71 x_{9} & -1.14 x_{5} & -8.57 x_{6} & -13.57 x_{4}\\
 x_{11}   &  5.0 & + 34.43 x_{1} & -7.43 x_{2} & +  0.43 x_{14} & -6.57 x_{9} & -3.29 x_{5} & -1.14 x_{6} & + 27.86 x_{4}\\
 x_{7}   &  6.66133814775e-16 & +  5.00 x_{1} & -1.00 x_{2} & +  0.00 x_{14} & -1.00 x_{9} & -1.00 x_{5} & +  1.00 x_{6} & +  5.00 x_{4}\\
 x_{12}   &  16.0 & -2.86 x_{1} & +  3.86 x_{2} & +  1.14 x_{14} & +  2.14 x_{9} & +  0.57 x_{5} & -12.71 x_{6} & -10.71 x_{4}\\
 x_{13}   &  7.0 & -57.57 x_{1} & + 21.57 x_{2} & +  0.43 x_{14} & + 10.43 x_{9} & + 10.71 x_{5} & -11.14 x_{6} & -49.14 x_{4}\\
 x_{3}   &  2.0 & -4.86 x_{1} & +  1.86 x_{2} & +  0.14 x_{14} & +  1.14 x_{9} & +  0.57 x_{5} & -1.71 x_{6} & -5.71 x_{4}\\
 x_{15}   &  36.0 & +  8.57 x_{1} & -19.57 x_{2} & -1.43 x_{14} & -4.43 x_{9} & +  4.29 x_{5} & +  8.14 x_{6} & + 29.14 x_{4}\\
\hline
z    &  -2.0 & + 19.86 x_{1} & -10.86 x_{2} & -0.14 x_{14} & -5.14 x_{9} & -3.57 x_{5} & + 10.71 x_{6} & + 22.71 x_{4}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  31.0 & + 65.71 x_{1} & -19.71 x_{2} & -1.29 x_{14} & -16.29 x_{9} & -17.14 x_{5} & + 19.43 x_{6} & + 80.43 x_{4}\\
 x_{10}   &  7.0 & -28.29 x_{1} & + 14.29 x_{2} & +  0.71 x_{14} & +  4.71 x_{9} & -1.14 x_{5} & -8.57 x_{6} & -13.57 x_{4}\\
 x_{11}   &  5.0 & + 34.43 x_{1} & -7.43 x_{2} & +  0.43 x_{14} & -6.57 x_{9} & -3.29 x_{5} & -1.14 x_{6} & + 27.86 x_{4}\\
 x_{7}   &  6.66133814775e-16 & +  5.00 x_{1} & -1.00 x_{2} & +  0.00 x_{14} & -1.00 x_{9} & -1.00 x_{5} & +  1.00 x_{6} & +  5.00 x_{4}\\
 x_{12}   &  16.0 & -2.86 x_{1} & +  3.86 x_{2} & +  1.14 x_{14} & +  2.14 x_{9} & +  0.57 x_{5} & -12.71 x_{6} & -10.71 x_{4}\\
 x_{13}   &  7.0 & -57.57 x_{1} & + 21.57 x_{2} & +  0.43 x_{14} & + 10.43 x_{9} & + 10.71 x_{5} & -11.14 x_{6} & -49.14 x_{4}\\
 x_{3}   &  2.0 & -4.86 x_{1} & +  1.86 x_{2} & +  0.14 x_{14} & +  1.14 x_{9} & +  0.57 x_{5} & -1.71 x_{6} & -5.71 x_{4}\\
 x_{15}   &  36.0 & +  8.57 x_{1} & -19.57 x_{2} & -1.43 x_{14} & -4.43 x_{9} & +  4.29 x_{5} & +  8.14 x_{6} & + 29.14 x_{4}\\
\hline
z    &  -2.0 & + 19.86 x_{1} & -10.86 x_{2} & -0.14 x_{14} & -5.14 x_{9} & -3.57 x_{5} & + 10.71 x_{6} & + 22.71 x_{4}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  38.9900744417 & -1.14 x_{13} & +  4.91 x_{2} & -0.80 x_{14} & -4.38 x_{9} & -4.91 x_{5} & +  6.71 x_{6} & + 24.33 x_{4}\\
 x_{10}   &  3.56079404467 & +  0.49 x_{13} & +  3.69 x_{2} & +  0.50 x_{14} & -0.41 x_{9} & -6.41 x_{5} & -3.10 x_{6} & + 10.57 x_{4}\\
 x_{11}   &  9.18610421836 & -0.60 x_{13} & +  5.47 x_{2} & +  0.68 x_{14} & -0.33 x_{9} & +  3.12 x_{5} & -7.81 x_{6} & -1.53 x_{4}\\
 x_{7}   &  0.60794044665 & -0.09 x_{13} & +  0.87 x_{2} & +  0.04 x_{14} & -0.09 x_{9} & -0.07 x_{5} & +  0.03 x_{6} & +  0.73 x_{4}\\
 x_{12}   &  15.6526054591 & +  0.05 x_{13} & +  2.79 x_{2} & +  1.12 x_{14} & +  1.63 x_{9} & +  0.04 x_{5} & -12.16 x_{6} & -8.28 x_{4}\\
 x_{1}   &  0.12158808933 & -0.02 x_{13} & +  0.37 x_{2} & +  0.01 x_{14} & +  0.18 x_{9} & +  0.19 x_{5} & -0.19 x_{6} & -0.85 x_{4}\\
 x_{3}   &  1.4094292804 & +  0.08 x_{13} & +  0.04 x_{2} & +  0.11 x_{14} & +  0.26 x_{9} & -0.33 x_{5} & -0.77 x_{6} & -1.57 x_{4}\\
 x_{15}   &  37.0421836228 & -0.15 x_{13} & -16.36 x_{2} & -1.36 x_{14} & -2.88 x_{9} & +  5.88 x_{5} & +  6.48 x_{6} & + 21.83 x_{4}\\
\hline
z    &  0.414392059553 & -0.34 x_{13} & -3.42 x_{2} & +  0.00 x_{14} & -1.55 x_{9} & +  0.12 x_{5} & +  6.87 x_{6} & +  5.76 x_{4}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  42.4563953488 & -1.64 x_{13} & + 15.59 x_{2} & -0.58 x_{14} & +  0.78 x_{9} & +  0.39 x_{5} & +  1.19 x_{6} & -28.51 x_{1}\\
 x_{10}   &  5.06686046512 & +  0.28 x_{13} & +  8.33 x_{2} & +  0.60 x_{14} & +  1.83 x_{9} & -4.10 x_{5} & -5.49 x_{6} & -12.39 x_{1}\\
 x_{11}   &  8.96802325581 & -0.57 x_{13} & +  4.80 x_{2} & +  0.67 x_{14} & -0.66 x_{9} & +  2.79 x_{5} & -7.46 x_{6} & +  1.79 x_{1}\\
 x_{7}   &  0.712209302326 & -0.10 x_{13} & +  1.19 x_{2} & +  0.04 x_{14} & +  0.06 x_{9} & +  0.09 x_{5} & -0.13 x_{6} & -0.86 x_{1}\\
 x_{12}   &  14.4738372093 & +  0.22 x_{13} & -0.85 x_{2} & +  1.05 x_{14} & -0.13 x_{9} & -1.76 x_{5} & -10.28 x_{6} & +  9.69 x_{1}\\
 x_{4}   &  0.142441860465 & -0.02 x_{13} & +  0.44 x_{2} & +  0.01 x_{14} & +  0.21 x_{9} & +  0.22 x_{5} & -0.23 x_{6} & -1.17 x_{1}\\
 x_{3}   &  1.18604651163 & +  0.12 x_{13} & -0.65 x_{2} & +  0.09 x_{14} & -0.07 x_{9} & -0.67 x_{5} & -0.42 x_{6} & +  1.84 x_{1}\\
 x_{15}   &  40.1511627907 & -0.59 x_{13} & -6.78 x_{2} & -1.17 x_{14} & +  1.76 x_{9} & + 10.64 x_{5} & +  1.53 x_{6} & -25.57 x_{1}\\
\hline
z    &  1.23546511628 & -0.46 x_{13} & -0.89 x_{2} & +  0.06 x_{14} & -0.32 x_{9} & +  1.38 x_{5} & +  5.56 x_{6} & -6.75 x_{1}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  42.9411764706 & -1.61 x_{13} & + 16.39 x_{2} & -0.53 x_{14} & +  0.96 x_{9} & -0.10 x_{10} & +  0.67 x_{6} & -29.69 x_{1}\\
 x_{5}   &  1.23529411765 & +  0.07 x_{13} & +  2.03 x_{2} & +  0.15 x_{14} & +  0.45 x_{9} & -0.24 x_{10} & -1.34 x_{6} & -3.02 x_{1}\\
 x_{11}   &  12.4117647059 & -0.38 x_{13} & + 10.46 x_{2} & +  1.08 x_{14} & +  0.59 x_{9} & -0.68 x_{10} & -11.19 x_{6} & -6.63 x_{1}\\
 x_{7}   &  0.823529411765 & -0.10 x_{13} & +  1.38 x_{2} & +  0.06 x_{14} & +  0.10 x_{9} & -0.02 x_{10} & -0.25 x_{6} & -1.13 x_{1}\\
 x_{12}   &  12.2941176471 & +  0.10 x_{13} & -4.43 x_{2} & +  0.79 x_{14} & -0.92 x_{9} & +  0.43 x_{10} & -7.92 x_{6} & + 15.02 x_{1}\\
 x_{4}   &  0.411764705882 & -0.01 x_{13} & +  0.88 x_{2} & +  0.04 x_{14} & +  0.31 x_{9} & -0.05 x_{10} & -0.52 x_{6} & -1.83 x_{1}\\
 x_{3}   &  0.352941176471 & +  0.07 x_{13} & -2.02 x_{2} & -0.00 x_{14} & -0.37 x_{9} & +  0.16 x_{10} & +  0.48 x_{6} & +  3.87 x_{1}\\
 x_{15}   &  53.2941176471 & +  0.12 x_{13} & + 14.82 x_{2} & +  0.37 x_{14} & +  6.51 x_{9} & -2.59 x_{10} & -12.72 x_{6} & -57.70 x_{1}\\
\hline
z    &  2.94117647059 & -0.37 x_{13} & +  1.92 x_{2} & +  0.26 x_{14} & +  0.29 x_{9} & -0.34 x_{10} & +  3.71 x_{6} & -10.92 x_{1}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  45.803577692 & -1.04 x_{13} & -8.11 x_{3} & -0.57 x_{14} & -2.05 x_{9} & +  1.24 x_{10} & +  4.60 x_{6} & +  1.72 x_{1}\\
 x_{5}   &  1.58996843213 & +  0.14 x_{13} & -1.00 x_{3} & +  0.14 x_{14} & +  0.07 x_{9} & -0.08 x_{10} & -0.85 x_{6} & +  0.87 x_{1}\\
 x_{11}   &  14.2388635566 & -0.01 x_{13} & -5.18 x_{3} & +  1.05 x_{14} & -1.34 x_{9} & +  0.17 x_{10} & -8.68 x_{6} & + 13.43 x_{1}\\
 x_{7}   &  1.06418800421 & -0.05 x_{13} & -0.68 x_{3} & +  0.05 x_{14} & -0.15 x_{9} & +  0.09 x_{10} & +  0.08 x_{6} & +  1.51 x_{1}\\
 x_{12}   &  11.5205191161 & -0.06 x_{13} & +  2.19 x_{3} & +  0.80 x_{14} & -0.11 x_{9} & +  0.07 x_{10} & -8.98 x_{6} & +  6.53 x_{1}\\
 x_{4}   &  0.565766397755 & +  0.03 x_{13} & -0.44 x_{3} & +  0.04 x_{14} & +  0.15 x_{9} & +  0.02 x_{10} & -0.31 x_{6} & -0.14 x_{1}\\
 x_{2}   &  0.174675552438 & +  0.04 x_{13} & -0.49 x_{3} & -0.00 x_{14} & -0.18 x_{9} & +  0.08 x_{10} & +  0.24 x_{6} & +  1.92 x_{1}\\
 x_{15}   &  55.8835496317 & +  0.64 x_{13} & -7.34 x_{3} & +  0.33 x_{14} & +  3.79 x_{9} & -1.39 x_{10} & -9.16 x_{6} & -29.28 x_{1}\\
\hline
z    &  3.27604349351 & -0.30 x_{13} & -0.95 x_{3} & +  0.25 x_{14} & -0.06 x_{9} & -0.18 x_{10} & +  4.17 x_{6} & -7.25 x_{1}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  51.6994494983 & -1.06 x_{13} & -6.99 x_{3} & -0.16 x_{14} & -2.11 x_{9} & +  1.27 x_{10} & -0.51 x_{12} & +  5.07 x_{1}\\
 x_{5}   &  0.496974192793 & +  0.14 x_{13} & -1.21 x_{3} & +  0.06 x_{14} & +  0.08 x_{9} & -0.09 x_{10} & +  0.09 x_{12} & +  0.25 x_{1}\\
 x_{11}   &  3.10291648772 & +  0.04 x_{13} & -7.30 x_{3} & +  0.27 x_{14} & -1.23 x_{9} & +  0.10 x_{10} & +  0.97 x_{12} & +  7.11 x_{1}\\
 x_{7}   &  1.16179283957 & -0.05 x_{13} & -0.66 x_{3} & +  0.06 x_{14} & -0.15 x_{9} & +  0.09 x_{10} & -0.01 x_{12} & +  1.57 x_{1}\\
 x_{6}   &  1.28235661578 & -0.01 x_{13} & +  0.24 x_{3} & +  0.09 x_{14} & -0.01 x_{9} & +  0.01 x_{10} & -0.11 x_{12} & +  0.73 x_{1}\\
 x_{4}   &  0.171748721352 & +  0.03 x_{13} & -0.51 x_{3} & +  0.01 x_{14} & +  0.15 x_{9} & +  0.02 x_{10} & +  0.03 x_{12} & -0.36 x_{1}\\
 x_{2}   &  0.482333190177 & +  0.03 x_{13} & -0.44 x_{3} & +  0.02 x_{14} & -0.19 x_{9} & +  0.08 x_{10} & -0.03 x_{12} & +  2.09 x_{1}\\
 x_{15}   &  44.1372349979 & +  0.70 x_{13} & -9.57 x_{3} & -0.48 x_{14} & +  3.90 x_{9} & -1.46 x_{10} & +  1.02 x_{12} & -35.94 x_{1}\\
\hline
z    &  8.62901651505 & -0.33 x_{13} & +  0.07 x_{3} & +  0.62 x_{14} & -0.11 x_{9} & -0.15 x_{10} & -0.46 x_{12} & -4.21 x_{1}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  49.3520158827 & -1.44 x_{13} & + 13.67 x_{4} & -0.30 x_{14} & -4.18 x_{9} & +  1.05 x_{10} & -0.98 x_{12} & + 10.03 x_{1}\\
 x_{5}   &  0.0895693341478 & +  0.08 x_{13} & +  2.37 x_{4} & +  0.04 x_{14} & -0.27 x_{9} & -0.12 x_{10} & +  0.01 x_{12} & +  1.11 x_{1}\\
 x_{11}   &  0.652336591326 & -0.35 x_{13} & + 14.27 x_{4} & +  0.12 x_{14} & -3.39 x_{9} & -0.13 x_{10} & +  0.48 x_{12} & + 12.29 x_{1}\\
 x_{7}   &  0.938989004276 & -0.08 x_{13} & +  1.30 x_{4} & +  0.05 x_{14} & -0.35 x_{9} & +  0.07 x_{10} & -0.05 x_{12} & +  2.04 x_{1}\\
 x_{6}   &  1.36430971289 & +  0.01 x_{13} & -0.48 x_{4} & +  0.09 x_{14} & +  0.06 x_{9} & +  0.02 x_{10} & -0.09 x_{12} & +  0.55 x_{1}\\
 x_{3}   &  0.335904092853 & +  0.05 x_{13} & -1.96 x_{4} & +  0.02 x_{14} & +  0.30 x_{9} & +  0.03 x_{10} & +  0.07 x_{12} & -0.71 x_{1}\\
 x_{2}   &  0.335751374465 & +  0.01 x_{13} & +  0.85 x_{4} & +  0.01 x_{14} & -0.32 x_{9} & +  0.07 x_{10} & -0.06 x_{12} & +  2.40 x_{1}\\
 x_{15}   &  40.9221136225 & +  0.19 x_{13} & + 18.72 x_{4} & -0.69 x_{14} & +  1.07 x_{9} & -1.76 x_{10} & +  0.38 x_{12} & -29.14 x_{1}\\
\hline
z    &  8.65241295052 & -0.32 x_{13} & -0.14 x_{4} & +  0.63 x_{14} & -0.09 x_{9} & -0.15 x_{10} & -0.46 x_{12} & -4.26 x_{1}\\
\end{array}\]


 $ x_{14} $ enters and $ x_{15} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  31.2788418708 & -1.52 x_{13} & +  5.40 x_{4} & +  0.44 x_{15} & -4.65 x_{9} & +  1.83 x_{10} & -1.15 x_{12} & + 22.90 x_{1}\\
 x_{5}   &  2.39086859688 & +  0.09 x_{13} & +  3.42 x_{4} & -0.06 x_{15} & -0.21 x_{9} & -0.22 x_{10} & +  0.04 x_{12} & -0.52 x_{1}\\
 x_{11}   &  7.84788418708 & -0.31 x_{13} & + 17.56 x_{4} & -0.18 x_{15} & -3.20 x_{9} & -0.44 x_{10} & +  0.55 x_{12} & +  7.17 x_{1}\\
 x_{7}   &  3.69599109131 & -0.07 x_{13} & +  2.56 x_{4} & -0.07 x_{15} & -0.28 x_{9} & -0.05 x_{10} & -0.03 x_{12} & +  0.07 x_{1}\\
 x_{6}   &  7.01046770601 & +  0.03 x_{13} & +  2.11 x_{4} & -0.14 x_{15} & +  0.21 x_{9} & -0.23 x_{10} & -0.04 x_{12} & -3.47 x_{1}\\
 x_{3}   &  1.58908685969 & +  0.06 x_{13} & -1.38 x_{4} & -0.03 x_{15} & +  0.33 x_{9} & -0.02 x_{10} & +  0.08 x_{12} & -1.60 x_{1}\\
 x_{2}   &  0.923608017817 & +  0.01 x_{13} & +  1.12 x_{4} & -0.01 x_{15} & -0.30 x_{9} & +  0.04 x_{10} & -0.05 x_{12} & +  1.98 x_{1}\\
 x_{14}   &  59.6788418708 & +  0.28 x_{13} & + 27.30 x_{4} & -1.46 x_{15} & +  1.55 x_{9} & -2.57 x_{10} & +  0.55 x_{12} & -42.50 x_{1}\\
\hline
z    &  46.0200445434 & -0.15 x_{13} & + 16.96 x_{4} & -0.91 x_{15} & +  0.89 x_{9} & -1.76 x_{10} & -0.11 x_{12} & -30.87 x_{1}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  37.4859444221 & -1.29 x_{13} & -3.91 x_{3} & +  0.32 x_{15} & -3.36 x_{9} & +  1.74 x_{10} & -0.84 x_{12} & + 16.64 x_{1}\\
 x_{5}   &  6.3274265002 & +  0.24 x_{13} & -2.48 x_{3} & -0.13 x_{15} & +  0.60 x_{9} & -0.28 x_{10} & +  0.23 x_{12} & -4.49 x_{1}\\
 x_{11}   &  28.0316552557 & +  0.44 x_{13} & -12.70 x_{3} & -0.56 x_{15} & +  0.97 x_{9} & -0.72 x_{10} & +  1.54 x_{12} & -13.18 x_{1}\\
 x_{7}   &  6.63672976238 & +  0.04 x_{13} & -1.85 x_{3} & -0.12 x_{15} & +  0.33 x_{9} & -0.09 x_{10} & +  0.12 x_{12} & -2.89 x_{1}\\
 x_{6}   &  9.43076923077 & +  0.12 x_{13} & -1.52 x_{3} & -0.18 x_{15} & +  0.71 x_{9} & -0.26 x_{10} & +  0.08 x_{12} & -5.91 x_{1}\\
 x_{4}   &  1.149416029 & +  0.04 x_{13} & -0.72 x_{3} & -0.02 x_{15} & +  0.24 x_{9} & -0.02 x_{10} & +  0.06 x_{12} & -1.16 x_{1}\\
 x_{2}   &  2.21369311317 & +  0.06 x_{13} & -0.81 x_{3} & -0.04 x_{15} & -0.03 x_{9} & +  0.03 x_{10} & +  0.01 x_{12} & +  0.68 x_{1}\\
 x_{14}   &  91.0581554571 & +  1.45 x_{13} & -19.75 x_{3} & -2.06 x_{15} & +  8.04 x_{9} & -3.01 x_{10} & +  2.10 x_{12} & -74.14 x_{1}\\
\hline
z    &  65.5114780507 & +  0.57 x_{13} & -12.27 x_{3} & -1.29 x_{15} & +  4.92 x_{9} & -2.03 x_{10} & +  0.85 x_{12} & -50.53 x_{1}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  11.144615532 & -0.38 x_{13} & -1.16 x_{3} & +  0.10 x_{15} & -0.30 x_{8} & +  0.52 x_{10} & -0.25 x_{12} & +  4.95 x_{1}\\
 x_{5}   &  13.0016283915 & +  0.01 x_{13} & -3.17 x_{3} & -0.07 x_{15} & -0.18 x_{8} & +  0.03 x_{10} & +  0.08 x_{12} & -1.53 x_{1}\\
 x_{11}   &  38.8180990924 & +  0.07 x_{13} & -13.83 x_{3} & -0.47 x_{15} & -0.29 x_{8} & -0.22 x_{10} & +  1.30 x_{12} & -8.40 x_{1}\\
 x_{7}   &  10.321679159 & -0.09 x_{13} & -2.23 x_{3} & -0.09 x_{15} & -0.10 x_{8} & +  0.08 x_{10} & +  0.04 x_{12} & -1.26 x_{1}\\
 x_{6}   &  17.3177279149 & -0.15 x_{13} & -2.34 x_{3} & -0.12 x_{15} & -0.21 x_{8} & +  0.11 x_{10} & -0.10 x_{12} & -2.41 x_{1}\\
 x_{4}   &  3.79755262339 & -0.05 x_{13} & -1.00 x_{3} & +  0.00 x_{15} & -0.07 x_{8} & +  0.11 x_{10} & -0.00 x_{12} & +  0.02 x_{1}\\
 x_{2}   &  1.83756794942 & +  0.07 x_{13} & -0.77 x_{3} & -0.04 x_{15} & +  0.01 x_{8} & +  0.01 x_{10} & +  0.02 x_{12} & +  0.51 x_{1}\\
 x_{14}   &  180.66301875 & -1.64 x_{13} & -29.08 x_{3} & -1.29 x_{15} & -2.39 x_{8} & +  1.16 x_{10} & +  0.10 x_{12} & -34.37 x_{1}\\
\hline
z    &  120.296486985 & -1.31 x_{13} & -17.97 x_{3} & -0.82 x_{15} & -1.46 x_{8} & +  0.52 x_{10} & -0.38 x_{12} & -26.21 x_{1}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  103.336003511 & -0.23 x_{13} & -34.00 x_{3} & -1.03 x_{15} & -0.98 x_{8} & -2.37 x_{11} & +  2.84 x_{12} & -14.99 x_{1}\\
 x_{5}   &  18.7819598376 & +  0.02 x_{13} & -5.23 x_{3} & -0.15 x_{15} & -0.22 x_{8} & -0.15 x_{11} & +  0.27 x_{12} & -2.78 x_{1}\\
 x_{10}   &  177.878305717 & +  0.30 x_{13} & -63.35 x_{3} & -2.16 x_{15} & -1.32 x_{8} & -4.58 x_{11} & +  5.96 x_{12} & -38.47 x_{1}\\
 x_{7}   &  24.7575990343 & -0.06 x_{13} & -7.38 x_{3} & -0.27 x_{15} & -0.21 x_{8} & -0.37 x_{11} & +  0.52 x_{12} & -4.38 x_{1}\\
 x_{6}   &  36.0388456052 & -0.12 x_{13} & -9.01 x_{3} & -0.34 x_{15} & -0.35 x_{8} & -0.48 x_{11} & +  0.53 x_{12} & -6.46 x_{1}\\
 x_{4}   &  22.8381433118 & -0.02 x_{13} & -7.78 x_{3} & -0.23 x_{15} & -0.21 x_{8} & -0.49 x_{11} & +  0.64 x_{12} & -4.10 x_{1}\\
 x_{2}   &  3.35399978053 & +  0.08 x_{13} & -1.31 x_{3} & -0.06 x_{15} & -0.00 x_{8} & -0.04 x_{11} & +  0.07 x_{12} & +  0.19 x_{1}\\
 x_{14}   &  386.574015143 & -1.28 x_{13} & -102.42 x_{3} & -3.80 x_{15} & -3.92 x_{8} & -5.30 x_{11} & +  7.00 x_{12} & -78.90 x_{1}\\
\hline
z    &  212.722155163 & -1.15 x_{13} & -50.89 x_{3} & -1.94 x_{15} & -2.15 x_{8} & -2.38 x_{11} & +  2.72 x_{12} & -46.20 x_{1}\\
\end{array}\]


 $ x_{12} $ enters and Unbounded Dictionary!


 $ x_{12} $ enters and Unbounded Dictionary!
\end{document}
